In my experience, chatting with a dragon isn't particularly useful.
You can use a stethoscope to check on a pet dragon, if you
can find one, but in my experience pet dragons generally can
take care of themselves pretty well as long as you keep them
with you all the time and continue exploring downward
through the mines.
They're more robust than dogs, so you shouldn't need to
provide them with healing.
They shouldn't go ferral unless you leave them alone (or
maybe if you use them as a steed repeatedly, if you're not a
knight; but dragons are kind of slow for steeds anyway).
Dragons generally only have hunger issues if you start doing
something time-consuming on a cleared level (polypiling,
extensive alchemy, sacrifests, any kind of farming, etc.) or
traveling back and forth across multiple cleared levels
(e.g., from deep in the main dungeon to Minetown and back).
So don't do that until you can stand to lose the pets (or
you have a stethoscope).
Eventually you will need to do some of those things, and
your dragons will go ferral; but by then you can probably
get along without powerful pets (barring certain kinds of
conduct play, notably pacifism). Former-pet dragons (of
most colors) are not especially dangerous once you have
reflection, just stay out of melee range, which is easy to
do unless you're polyselfed into something painfully slow.